Open Research Database v2.3 RamanujanTools pFq

Conservative Matrix Field Atlas

A public mathematical database of Conservative Matrix Fields — path-independent matrix recurrences connecting continued fractions, pFq hypergeometric series, and named constants such as ζ(3), π, ln(2) and ζ(5).

BETA

The CMF Atlas is currently in its beta stage. This platform is designed to facilitate mathematical research by making Conservative Matrix Fields (CMF) more accessible to the global community. We are actively verifying existing CMFs in the database and searching for new high dimensional CMFs.

CMFs by Matrix Size

CMFs by Parameter Dimension

Highlighted Families

2D · Apéry ζ(3) A-certified
\(f(x,y) = (x+y)^3\)

The cubic symmetric CMF whose walk limit is ζ(3). Foundational to Apéry's irrationality proof.

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pFq · Higher-dim A-certified
\({}_pF_q(a_1,\ldots;\,b_1,\ldots;\,z)\)

Hypergeometric CMFs from RamanujanTools — 3F2, 4F3, 5F4, 6F5 families targeting ζ(3), ζ(5), π, Catalan.

Browse pFq CMFs →
Interactive Explorer
Numerical convergence

Walk any CMF with explicit f(x,y) up to depth 2000 using 30-digit mpmath precision. Compare against known constants from the registry.

Open Explorer →

Notable Entries

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ID Dim f(x,y) / Type Constant Cert
#1 2D \((x+y)^3\) ζ(3) A View →
#2 2D \((x+y)^3 + (x+y)\) ζ(3) A View →
#5594 5D pFq hypergeometric ζ(3) A+ View →

Cite CMF Atlas

Vesterlund, D. (2026). CMF Atlas: A Database of Conservative Matrix Fields. https://davidvesterlund.com/cmf-atlas/